| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw histogram then estimate mean/standard deviation |
| Difficulty | Easy -1.2 This is a straightforward statistics question requiring basic skills: simple arithmetic to find a missing frequency, calculating an estimated mean from grouped data using midpoints, and drawing a histogram with unequal class widths (requiring frequency density). All techniques are standard S1 procedures with no conceptual challenges or problem-solving required. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| \(0 < t \leqslant 1\) | \(1 < t \leqslant 2\) | \(2 < t \leqslant 5\) | \(5 < t \leqslant 10\) | \(10 < t \leqslant 30\) | ||
| Frequency | 14 | 46 | 102 | \(a\) | 40 |
| Answer | Marks |
|---|---|
| \(a = 40\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Mean} = \frac{0.5\times14+1.5\times46+3.5\times102+7.5\times their\ 40+20\times40}{242}\) | M1 | Numerator: 5 products with at least 3 acceptable mid-points \(\times\) appropriate frequency FT (i). Denominator: 242 CAO. \(\frac{1533}{242}\) implies M1, but if FT an unsimplified expression required |
| \(= \frac{1533}{242}\) | ||
| \(= 6\frac{81}{242}\) or \(6.33\) | A1 | CAO (6.3347… rounded to 3 or more SF) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{fd} = 14, 46, 34, \left(\frac{their\ (i)}{5}=\right) 8, 2\) | M1 | Attempt at fd [f/(attempt at cw)] or scaled freq |
| Correct heights on diagram | A1FT | Correct heights seen on diagram with linear vertical scale from \((x, 0)\). FT their \(\frac{a}{5}\) only |
| Correct bar widths | B1 | Correct bar widths (1:1:3:5:20) at axis, visually no gaps, with linear horizontal scale from \((0, y)\), first bar starting at \((0,0)\) |
| Labels and scale | B1 | Labels (time, mins, and fd(OE) seen, some may be as a title) and a linear scale with at least 3 values marked on each axis. (Interval notation not acceptable) |
## Question 5(i):
$a = 40$ | B1 |
---
## Question 5(ii):
$\text{Mean} = \frac{0.5\times14+1.5\times46+3.5\times102+7.5\times their\ 40+20\times40}{242}$ | M1 | Numerator: 5 products with at least 3 acceptable mid-points $\times$ appropriate frequency FT (i). Denominator: 242 CAO. $\frac{1533}{242}$ implies M1, but if FT an unsimplified expression required
$= \frac{1533}{242}$ | |
$= 6\frac{81}{242}$ or $6.33$ | A1 | CAO (6.3347… rounded to 3 or more SF)
---
## Question 5(iii):
$\text{fd} = 14, 46, 34, \left(\frac{their\ (i)}{5}=\right) 8, 2$ | M1 | Attempt at fd [f/(attempt at cw)] or scaled freq
Correct heights on diagram | A1FT | Correct heights seen on diagram with linear vertical scale from $(x, 0)$. FT their $\frac{a}{5}$ only
Correct bar widths | B1 | Correct bar widths (1:1:3:5:20) at axis, visually no gaps, with linear horizontal scale from $(0, y)$, first bar starting at $(0,0)$
Labels and scale | B1 | Labels (time, mins, and fd(OE) seen, some may be as a title) and a linear scale with at least 3 values marked on each axis. (Interval notation not acceptable)
---5 The lengths, $t$ minutes, of 242 phone calls made by a family over a period of 1 week are summarised in the frequency table below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
\begin{tabular}{ l }
Length of phone \\
call $( t$ minutes $)$ \\
\end{tabular} & $0 < t \leqslant 1$ & $1 < t \leqslant 2$ & $2 < t \leqslant 5$ & $5 < t \leqslant 10$ & $10 < t \leqslant 30$ \\
\hline
Frequency & 14 & 46 & 102 & $a$ & 40 \\
\hline
\end{tabular}
\end{center}
(i) Find the value of $a$.\\
(ii) Calculate an estimate of the mean length of these phone calls.\\
(iii) On the grid, draw a histogram to illustrate the data in the table.\\
\includegraphics[max width=\textwidth, alt={}, center]{a813e127-d116-411c-88ec-2443fdbc9391-07_2002_1513_486_356}
\hfill \mbox{\textit{CAIE S1 2018 Q5 [7]}}